Albert and Bousquet-Mélou's $(x+1)$-positivity conjecture for balanced path projections

Let $\mathcal{V}$ be a positive vertical path with $u$ up-steps and $d$ down-steps, let $Q$ be the set of quarter-planar paths with $r$ right-steps, $l$ left-steps, and vertical projection $\mathcal{V}$, and let $G_1$ be the peak-count generating function over $Q$. A polynomial is $(x+1)$-positive when it belongs to $\mathbb{N}_0[x+1]$. Albert and Bousquet-Mélou's conjecture. If $a=b$ and $c=d$, then $G_1$ is $(x+1)$-positive. The statement is presented as Albert and Bousquet-Mélou's Conjecture (P2), and the paper says that it implies their quarter-planar-loop conjecture; the balanced parameters and the roles of $a,b,c,d$ should be checked against the paper's earlier definitions.